Unique continuation and the regularity of elliptic PDEs and generalized minimal submanifolds

椭圆偏微分方程和广义最小子流形的唯一延拓和正则性

• 批准号: 2350351
• 负责人: Zihui Zhao
• 金额: $ 25.37万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350351&HistoricalAwards=false
• 关键词: Unique; continuation; regularity; elliptic; PDEs

This award supports research on the regularity of solutions to elliptic partial differential equations and regularity of generalized minimal submanifolds. Elliptic differential equations govern the equilibrium configurations of various physical phenomena, for instance, those arising from minimization problems for natural energy functionals. Examples include the shape of free-hanging bridges, the shape of soap bubbles, and the sound of drums. Elliptic differential equations are also used to quantify the degree to which physical objects are bent or distorted, with far-reaching implications and applications in geometry and topology. The proposed research focuses on the regularity of solutions to such equations. Questions to be addressed include the following: Do non-smooth points (singularities) exist? How large can the set of singularities be? What is the behavior of the solution near a singularity? Is it possible to perturb the underlying environment in order to eliminate the singularity? The project will also provide opportunities for the professional development of graduate students, both via individual mentoring and via the organization of a directed learning seminar on geometric analysis and geometric measure theory.The mathematical objectives of the project are twofold. First, the principal investigator will study unique continuation for solutions to elliptic partial differential equations, with a focus on quantitative estimates on the size and structure of the singular set of these solutions. A second topic for consideration is the regularity theory for generalized minimal submanifolds (a generalized notion of smooth submanifolds which arise as critical points for the area functional under local deformations). In particular, the principal investigator will study branch singular points in the interior as well as at the boundary of a generalized minimal submanifold, under an area-minimizing or stability assumption. Research on the latter topic, which can be viewed as a non-linear analogue of quantitative unique continuation for elliptic equations, requires the integration of ideas from geometric measure theory, partial differential equations and geometric analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持对椭圆型偏微分方程解的正则性和广义极小子流形正则性的研究。椭圆型微分方程控制着各种物理现象的平衡构型，例如，由自然能量泛函的最小化问题引起的那些现象。例如，自由悬挂桥的形状、肥皂泡的形状和鼓声。椭圆型微分方程也被用来量化物理对象弯曲或扭曲的程度，在几何学和拓扑学中有着深远的影响和应用。本文的研究重点是这类方程解的正则性。需要解决的问题包括：是否存在非光滑点(奇点)？奇点的集合能有多大？解在奇点附近的行为是什么？有没有可能为了消除奇点而扰乱底层环境？该项目还将通过个人辅导和组织几何分析和几何测量理论的定向学习研讨会，为研究生提供专业发展的机会。首先，主要研究者将研究椭圆型偏微分方程解的唯一延拓性，重点是对这些解的奇异集的大小和结构的定量估计。第二个要考虑的主题是广义极小子流形的正则性理论(在局部变形下作为面积泛函的临界点出现的光滑子流形的广义概念)。特别是，在面积最小化或稳定性的假设下，主要研究者将研究广义极小子流形内部和边界上的分支奇点。对后一个主题的研究，可以被视为椭圆型方程定量唯一延拓的非线性模拟，需要整合几何测度论、偏微分方程式和几何分析的思想。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。